The Lumer Phillips Theorem
The Hille-Yosida theorem provides a set of conditions on the operator A that are necessary
and sufficient in order for -A to generate a strongly continuous semigroup of contractions. It
is often more convenient to have an equivalent set of conditions as described in the
following theorem.
Theorem (Lumer-Phillips) Suppose A : D A  H is closed and densely defined. Then the
following are equivalent conditions:
−A generates a C 0 semigroup of contractions on H
A is accretive and I + A : D A  H is surjective
1.
2.
Proof- 2  1: Suppose A is accretive. Then for λ > 0
||λI + Ax|| H ≥ λ||x|| H
∀x ∈ D A
Letting z = λI + Ax this becomes,
||z|| H ≥ λ||λI + A −1 z|| H .
In particular, since I + A is surjective, this previous estimate implies
||I + A −1 || LH ≤ 1.
Now observe that if 0 < λ < 2, then formally,
λI + A −1 = I + A −1 I + λ − 1I + A −1  −1
∞
= I + A −1 ∑ n=0 1 − λI + A −1  n
Note that if 0 < λ < 2, then 1 − λI + A −1 ∈ LH and
||1 − λI + A −1 || LH ≤ |λ − 1|≤ 1.
Then the Neumann series for 1 − λI + A −1 converges and we get,
I + λ − 1I + A −1
−1
∞
= ∑ n=0 1 − λI + A −1  n
Now let B = I + λ − 1I + A −1  and note that
λ + AI + A −1 B −1 = λ − 1 + I + AI + A −1 B −1
= λ − 1I + A −1 + IB −1 = BB −1 = I
and
I + A −1 B −1 λ + A = I + A −1 B −1 λ − 1 + I + A
= I + A −1 B −1 λ − 1I + A −1 + II + A
= I + A −1 B −1 BI + A = I;
i.e.,
λ + A −1 = I + A −1 B −1
and
1
||λ + A −1 || LH = ||I + A −1 || LH ||B −1 || LH
∞
≤ ∑ n=0 1 − λ n = 1/λ
0 < λ < 2.
Now it follows that
μI + A −1 = λI + A −1 I + μ − λλI + A −1  −1
and for μ > 0 and |μ − λ|||λI + A −1 || LH < 1, we have
μI + A −1 exists and ||μI + A −1 || LH ≤ 1/μ
Repeating this process we see that λI + A −1 exists for all λ > 0, and
||λI + A −1 || LH ≤ 1/λ
Then the hypotheses of the Hille-Yosida theorem are satisfied and −A generates a C 0
semigroup of contractions on H. This proves (2) implies (1).
Proof that 1  2 : If −A generates a C 0 semigroup of contractions on H then
Stx − x, x H = Stx, x H − ||x|| 2H ≤ ||Stx|| H − ||x|| H ||x|| H ≤ 0
This implies that for all x in D A
Stx − x/t, x H  −Ax, x H ≤ 0
i.e., A is accretive. In addition, it follows from the Hille-Yosida theorem that λ + A is
surjective for all λ > 0, and for λ = 1 in particular.■
Examples1) Let H = H 0 R , A = ∂ x , D A = H 1 R. Note that u ∈ H 1 R implies that ux is
continuous and tends to zero as |x| tends to infinity.
Then
Au, u H = ∫ u ′ xuxdx = 1/2 ux 2 | x=∞
x=−∞ = 0
R
∀u ∈ D A
and A is accretive. Note that according to this, − A is also accretive. Next, note that
I ± Au = ux ± u ′ x = fx
u ∈ DA, f ∈ H
implies 1 ± iαUα = Fα where U, F denote Fourier transforms of u and f,
respectively. Then
Uα =
Fα
Fα
= 1 ∓ iα
= Gα ∓ iαGα
1 ± iα
1 + α2
and
ux = gx ∓ g ′ x
where
gx = T −1
F
Fα
1 + α2
= ∫ e |x−y| fydy.
R
This shows that I + A : D A  H is onto and now it follows from the Lumer-Phillips theorem
that −A generates S(t), a C 0 − s/g of contractions, on H. This implies, in turn that for each
2
u 0 ∈ D A there exists a unique ut satisfying,
u′t + Aut = ∂ t ux, t + ∂ x ux, t = 0,
u0 = u 0 .
Since it is evident that ux, t = u 0 x − t solves the initial value problem, it follows by
uniqueness that Stu 0 x = u 0 x − t.
Since −A is also accretive and I − A : D  H is onto, +A must also generate a C 0 − s/g
of contractions, Zt. Then the solution of
u′t − Aut = ∂ t ux, t − ∂ x ux, t = 0,
u0 = u 0 ∈ D A
is given by ux, t = Ztu 0 x = u 0 x + t = S−tu 0 x. Then uniqueness implies that
Zt = S−t. In this case, the operator A generates a C 0 group of contractions on H. That is,
Ut =
if
t≥0
Z−t if
t<0
St
satisfies
UtU−t = StS−t = S0 = I.
Then Ut : t ∈ R is a group of contractions on H since for each operator in the collection,
the inverse operator is also in the collection.
2) Let H = H 0 U , for U ⊂ R n , open and bounded. Let Au = −divM∇u, for M a
symmetric positive definite matrix, and let D A = H 10 U ∩ H 2 U. Then
Au, u H = Bu, u = ∫ ∇u ⋅ M∇udx ≥ 0
U
∀u ∈ D A .
This shows that A is accretive and, from previous results regarding elliptic operators, it is
known that A + λI is surjective for λ ≥ 0. For λ = 1, in particular, A + I is surjective and the
Lumer-Phillips theorem asserts that −A generates a C 0 − s/g of contractions, S(t). Then the
IVP
u ′ t + Aut = ft,
u0 = u 0
has a unique solution for all u 0 ∈ D A and for all f ∈ C 1 0. ∞ : H. The solution is given by
t
ut = Stu 0 + ∫ St − τfτdτ.
0
This abstract IVP stands for the problem
∂ t ux, t − divM∇ux, t = fx, t
ux, 0 = u 0 x
ux, t = 0
in U T
in U
on Γ × 0, T
and then
t
ux, t = ∑ n>0 e −λ n t u 0 , φ n  H φ n x + ∑ n>0 ∫ e −λ n t−τ f⋅, τ, φ n  H dτ φ n x,
0
where λ n , φ n  denote the eigenvalues and normalized eigenfunctions associated with the
elliptic operator A. Evidently
Stu 0 = ∑ n>0 e −λ n t u 0 , φ n  H φ n x.
3
Groups
A family Gt : t ∈ R ⊂ LH is said to form a group on H if:
i
ii
GtGs = Gt + s
G0 = I
∀s, t ∈ R
Note that GtG−t = G0 = I so that for each real t, Gt −1 = G−t. The group G(t) is
said to be a C 0 -group on H if, Gtx  x in H as t tends to 0, for all x in H. As in the case of
semigroups, there is a closed, densely defined linear operator associated with G(t),
Bx = lim h0
Gh − I
x
h
for all x ∈ D B = x ∈ H : lim h0
Gh − I
x exists
h
and we say this operator B is the generator for the group. Note that if G(t) is generated by B,
then B generates a C 0 -s/g, St = Gt for t ≥ 0, and −B generates a C 0 -s/g, S−t = Gt for
t ≤ 0. In this case, the Lumer-Phillips theorem implies that B is accretive and
I + B : D B  H is surjective and, in addition, − B is accretive and I − B : D B  H is
surjective. Note that when B and −B are both accretive, then
Bx, x H ≥ 0
and
−Bx, x H ≥ 0
Bx, x H = 0
∀x ∈ D B
∀x ∈ D B
i.e.,
In this case we say the operator B is conservative. Note that this implies
d/dt||Gtx|| 2H = 2G ′ tx, Gtx H = 2BGtx, Gtx H = 0
i.e.,
||Gtx|| 2H = ||x|| 2H
for all t ∈ R
Then ||Gt|| LH = 1 and we say that
Gt : t ∈ R
is a unitary group on H.
Example- Consider the IVP
∂ tt ux, t − ∂ xx ux, t = 0,
ux, 0 = fx,
∂ t ux, 0 = gx
u0, t = u1, t = 0,
u1 = ∂xu
u2 = ∂tu
Let
0 < x < 1, t > 0
t > 0,
0 < x < 1.
∂ t u 1 = ∂ xt u = ∂ x u 2
∂ t u 2 = ∂ tt u = ∂ x u 1
Then
∂t
u1
u2
−
0 1
1 0
∂x
u1
u2
=
0
0
and
u1
u2
t = 0 =
f ′ x
gx
i.e.,
4
∂ t Ut − AUt = 0,
U0 = U 0
where
H = L 2 0, 1 2 ,
A=
0 1
1 0
∂x
D A = U ∈ H : u 1 ∈ H 1 0, 1, u 2 ∈ H 10 0, 1
Then
1
1
0
0
AU, U H = ∫ ∂ x u 2 ⋅ u 1 + u 2 ⋅ ∂ x u 1 dx = ∫ d/dxu 1 u 2 dx
1
= u 1 u 2 | x=1
x=0 = 0 (since u 2 ∈ H 0 0, 1
This proves A is conservative. Now for λ ≠ 0, F ∈ H, consider
λ
u1
u2
+A
u1
u2
=
λu 1 + ∂ x u 2
λu 2 + ∂ x u 1
=
F1
F2
Then
λ∂ x u 1 + ∂ xx u 2 = ∂ x F 1
and
∂ x u 1 = F 2 − λu 2 ,
or
∂ xx u 2 − λu 2 = ∂ x F 1 − λF 2
Since ∂ x F 1 − λF 2 ∈ H −1 0, 1, this last equation has a unique weak solution u 2 ∈ H 10 0, 1, by
the previously developed elliptic theory. Then
λu 1 = F 1 − ∂ x u 2 ∈ L 2 0, 1,
∂ x u 1 = F 2 − λu 2 ∈ L 2 0, 1,
so u 1 ∈ H 1 0, 1 and U ∈ D A . This shows that λ + A : D A  H is surjective for all λ ≠ 0. ■
We have the following version of the Hille-Yosida theorem for groups rather than
semigroups.
Theorem The following statements are equivalent:
1.
2.
B : D B  H generates a C 0 − unitary group on H
a) B is closed and densely defined
b) ∀λ ≠ 0 λ − B : D A  H is one to one and onto
with
λ − B −1 LH ≤ 1
λ
Proof- Suppose B generates a group, {Gt : t ∈ R. Then B generates a contraction
semigroup, Gt : t ≥ 0 and −B also generates a contraction semigroup,
G−t : t ≥ 0. Then Hille-Yosida theorem implies that both B and − B satisfy the
necessary conditions for generating a contraction semigroup and this implies 2.
Conversely, suppose the conditions 2 hold. Then, again by the Hille-Yosida theorem,
B generates a contraction semigroup S + t : t ≥ 0 and − B generates a contraction
semigroup S − t : t ≥ 0, and these semigroups commute. For x 0 ∈ D B
5
d
dt
S + tS − tx 0  = 0
t≥0
from which it follows that S + tS − t = I t ≥ 0. Then
Gt =
S + t
t≥0
S − −t
t≤0
satisfies all of the conditions for a group. i.e., (i) and (ii) are evident and
1 = ||Gt G−t|| ≤ ||Gt|| ||G−t|| ≤ ||Gt|| ≤ 1.
Finally, to see that B generates G(t), note that B generates S + t : t ≥ 0 so for h > 0,
S + h − Ix
Ghx − x
=
 Bx.
h
h
and − B generates S − t : t ≥ 0, so for h < 0
Ghx − x
S −h − Ix
=− −
 −−Bx. ■
h
−h
We have also a version of the Lumer-Phillips theorem for groups.
Theorem The following statements are equivalent:
1.
2.
. B : D B  H generates a C 0 − unitary group on H
a) B is closed and densely defined
b) B is conservative
c) λ + B is onto for some λ > 0 and for some λ < 0.
Example- Consider the IVP for the wave equation
∂ tt ux, t − ∂ xx ux, t = 0
ux, 0 = fx,
∂ t ux, 0 = gx,
u0, t = u1, t = 0,
Let
0 < x < 1, 0 < t < T,
0 < x < 1,
0 < t < T.
u 1 x, t = ∂ x ux, t
u 2 x, t = ∂ t ux, t
so
∂ t u 1 x, t = ∂ xt ux, t = ∂ x u 2 x, t
∂ t u 2 x, t = ∂ tx ux, t = ∂ x u 1 x, t
u 1 x, 0 = f ′ x
u 2 x, 0 = gx;
i.e.,
⃗ t =
∂tU
0 1
1 0
⃗ t,
∂xU
⃗ x, 0 =
U
f′
g
.
Let
H = L 2 0, 1 2 ,
DA =
⃗ ∈ H : u 1 ∈ H 1 0, 1, u 2 ∈ H 10 0, 1
U
Then
⃗, U
⃗
AU
1
H
1 d
dx
= ∫ u 1 ∂ x u 2 + u 2 ∂ x u 1  dx = ∫
0
0
u 1 u 2 dx = u 1 u 2 | x=1
x=0 = 0
which shows that A is conservative and must therefore generate a group if we can show
6
⃗ =
that condition 2c of the previous theorem is satisfied. But for λ ≠ 0 and F
⃗ + AU
⃗ =F
⃗
λU
λu 1 + ∂ x u 2

λu 2 + ∂ x u 1
=
F1
F2
∈H
F1
F2
λ∂ x u 1 + ∂ xx u 2 = ∂ x F 1
But in this case,
∂ x u 1 = F 2 − λu 2
and
hence
λF 2 − λu 2  + ∂ xx u 2 = ∂ x F 1 ,
or
∂ xx u 2 − λ 2 u 2 = ∂ x F 1 − λF 1 .
This last equation is uniquely solvable with u 2 ∈ H 10 0, 1 since ∂ x F 1 − λF 1 ∈ H −1 0, 1.
Next, λu 1 + ∂ x u 2 = F 1 implies λu 1 = ∂ x F 1 − ∂ x u 2 ∈ L 2 0, 1.
In addition, ∂ x u 1 = F 2 − λu 2 ∈ L 2 0, 1 hence u 1 ∈ H 1 0, 1 and
⃗ =
U
u1
u2
∈
H 1 0, 1
H 10 0, 1
= DA.
This shows that λ + A : D A  H is onto for λ ≠ 0.
Then A generates a group which we know from previous experience is given by
⃗ x, 0 =
GtU
1
2
1
2
f̃ ′ x + t + ̃f ′ x − t +
f̃ ′ x + t − ̃f ′ x − t +
1
2
1
2
g̃ x + t − g̃ x − t
g̃ x + t + g̃ x − t
where ̃f, g̃ denote the odd 2-periodic extensions of f and g.
7